Solving Equations of Free Vibration for a Cylindrical Shell Rotating on Rollers by the Fourier Method

摘要

The small free vibrations of an infinite circular cylindrical shell rotating about its axis at a constant angular velocity are considered. The shell is supported on n absolutely rigid cylindrical rollers equispaced on its circle. The roller-supported shell is a model of an ore benefication centrifugal concentrator with a floating bed. The set of linear differential equations of vibrations is sought in the form of a truncated Fourier series containing N terms along the circumferential coordinate. A system of 2 N – n linear homogeneous algebraic equations with 2 N – n unknowns is derived for the approximate estimation of vibration frequencies and mode shapes. The frequencies ω~( k ), k = 1, 2, …, 2 N – n , are positive roots of the (2 N – n )th-order algebraic equation D (ω_(2)) = 0, where D is the determinant of this set. It is shown that the system of 2 N – n equations is equivalent to several independent systems with a smaller number of unknowns. As a consequence, the (2 N – n )th-order determinant D can be written as a product of lower-order determinants. In particular, the frequencies at N = n are the roots of algebraic equations of an order is lower than 2 and can be found in an explicit form. Some frequency estimation algorithms have been developed for the case of N > n . When N increases, the number of found frequencies also grows, and the frequencies determined at N = n are refined. However, in most cases, the vibration frequencies can not be found for N > n in an explicit form.